summing up the two contributions to the total polarization. This follows from Eq. (2.18) successively substituting in it p = PudV and p = PedV, α0 = αudV and α0 = αedV. We have α0 = p/E = PedV/E = (Pe/E)dV = αedV. The same for αu. We determine now the contributions arising from the various interactions among different volume elements. To this aim, we consider the contribution to the whole field strength E given by the fields: Ec, arising from the charges in vacuum, Eu, arising from the U-type polarization and Ee from the E-type, that is,
We see from Eq. (2.19) that the free energy of interaction of the entire field with the dipole of E-type polarization in a volume dV is:
Similarly, for the U-type polarization in the same volume dV we have the interaction energy:
Finally, one should consider the free energy of interaction among charges. This term is the same as that among charges in vacuum because the terms describing the interactions among charges and polarizations have already been considered. We have, for unit volume [12, p. 288]:
The expression for the electrostatic free energy F of the system will be the sum of the earlier terms integrated over the whole volume of the system. In performing the integration, some terms must be divided by two in order not to count the interactions twice. Let us consider, for example, the integral
The differential term Pe(r) · Ee(r)dV is the interaction between the dipole Pe(r)dV and the electric field Ee(r) which is the resultant in r of the electric field strengths due to all the electronically polarized volume elements dV′ ≠ dV. At the same time, the dipole Pe(r)dV will participate in the electric field Eeon all dV′ ≠ dV. This means that the interaction between two dipole elements, Pe(r1)dV1 and Pe(r2)dV2 say, is counted twice in the integral, which is then to be divided by two in order to correctly take into account the interactions among different electronically polarized volume elements.
The terms which are to be divided by a factor of two are those in the integral:
Marcus thus obtained:
Using Eq. (2.20) and the relations Pe = αeE, P = Pe + Pu into Eq. (2.21), he finally obtained:
which is identical with Eq. (2.14).
2.5.Restraints Imposed on the Forms X∗ and X of the Transition State
In the framework of the 1956 theory, we have seen that the two forms X∗ and X of the TS have the following properties:
(i)Same atomic configuration
(ii)Same total electronic energy
(iii)Same U-type polarization, that is,
Moreover, the transition X∗ → X is of a Franck–Condon type [16] and so
(iv)The momentum distribution of the atoms in X∗ and X is the same
Properties (i) and (iv) say that the configurational and the thermal entropy [17] of X∗ and X are the same so that the only possible entropy difference between them can arise from a possible difference in electronic degeneracy between products and reactants, denoted by M. as ΔSe. Designating with ∗ the product of the electronic degeneracies of the reactants and with the same for the products, ΔSe is given by [1, 18]:
ΔSe is usually equal or close to zero [1].
Since F∗ = U∗ − TS∗, F = U − TS and U∗ = U for X∗ and X, we have then F − F∗ = −T ΔSe.
In his first treatment of the kinetics of ET reactions, M. considers a further restriction on the reactants: all atoms within the sphere of radius a “maintain their relative positions throughout the reaction” so that consequently the radii a1 and a2 remain fixed throughout the reaction, as seen earlier.
Because of all of the earlier restrictions, the overall standard free energy of formation of the products from the reactants ΔF0 can be written as sum of three terms:
(i)
(ii)−T ΔSe, free energy of formation of X state from X∗
(iii)
This equation “summarizes the restraints imposed upon the two intermediate states.”
2.6.Minimization of the Free Energy Subject to the Free Energy Restriction (2.24)